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We describe the Poisson ideals and attached symplectic geometry of a clusteralgebra with compatible Poisson structure. We apply these results to determinethe spectrum of a quantum cluster algebra. As an application, we describe thetopology on the spectra of quantized coordinate rings such as quantum matricesand the quantized function algebra of the general linear group.

Poisson Ideals in Cluster Algebras and the Spectra of Quantized Coordinate Rings